# Recent Developments in the Theory of Lorentz Spaces and Weighted Inequalities

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*María J. Carro; José A. Raposo; Javier Soria*

The main objective of this work is to bring together
two well known and, a priori, unrelated theories dealing with weighted
inequalities for the Hardy-Littlewood maximal operator \(M\). For this,
the authors consider the boundedness of \(M\) in the weighted Lorentz
space \(\Lambda^p_u(w)\). Two examples are historically relevant as a
motivation: If \(w=1\), this corresponds to the study of the
boundedness of \(M\) on \(L^p(u)\), which was characterized by
B. Muckenhoupt in 1972, and the solution is given by the so called
\(A_p\) weights. The second case is when we take \(u=1\). This is
a more recent theory, and was completely solved by M.A. Ariño and B.
Muckenhoupt in 1991. It turns out that the boundedness of \(M\)
on \(\Lambda^p(w)\) can be seen to be equivalent to the boundedness of
the Hardy operator \(A\) restricted to decreasing functions of
\(L^p(w)\), since the nonincreasing rearrangement of \(Mf\) is pointwise
equivalent to \(Af^*\). The class of weights satisfying this
boundedness is known as \(B_p\).

Even though the \(A_p\) and \(B_p\) classes enjoy some
similar features, they come from very different theories, and so are the
techniques used on each case: Calderón–Zygmund decompositions and
covering lemmas for \(A_p\), rearrangement invariant properties and
positive integral operators for \(B_p\).

This work aims to give a unified version of these two theories. Contrary to
what one could expect, the solution is not given in terms of the limiting cases
above considered (i.e., \(u=1\) and \(w=1\)), but in a rather
more complicated condition, which reflects the difficulty of estimating the
distribution function of the Hardy-Littlewood maximal operator with respect to
general measures.